People have thought, people have wondered about infinity for thousands of years.
We're going to continue that tradition of thinking about infinity, but we're going
to focus in on one specific instance of infinity in mathematics.
What does it mean to say that the limit of a function equals infinity?
Here's what it means. The limit of F of X, as X approaches A
equals infinity means that F of X is as large as you like.
Provided x is close enough to A. As a bit of a warning, I'm not saying
that this number limit F of X, X approaches A is equal to this number,
infinity is not a number. What I am doing here is attaching a
precise meaning to the entire statement that the limit of F of X is infinity.
So, that's what it means precisely, but what does it mean practically.
Let's go to the board. This is an example,
the limit of one over X squared as X approaches zero.
Now this limit is not equal to a number. One over X squared is not getting close
to a number when X is approaches zero. But one over X squared is as big as I
want it to be if I'm willing to put X close to zero.
For instance, if X is within a 1000th of zero, what happens?
Well to say X is within a 1000th of zero is to say that X is between minus.00
and.001. Now, X is also non zero here because I
don't want to divide by zero. So I've got a non zero number within a
1000th of zero. What does that mean about X squared?
[SOUND]. Well that means that X squared is, it's
bigger than zero because if I square any non zero number, it's positive and it's
less than, [SOUND] a millionth. So if I take a number within a thousandth
of zero and I square it, that was within a millionth of zero.
And [SOUND] what does that mean about one over X squared?
Well if X squared is within a millionth of zero, one over X squared is now
bigger, [SOUND] than a million. What happened, right?
I made one over x squared very large by making x close enough but not equal to
zero. And there's nothing particularly special
about these numbers. If I wanted one over x squared to be
bigger than a trillion, I would just need x to be within a millionth of zero.
This example is at once maybe too complicated because I've got some
explicit numbers in here. And maybe this formula is not complicated
enough to, to really get a sense of what's going on.
Let's look at a slightly more complicated function.
So what's the limit of X plus two over X squared minus 2x plus one as X approaches
one. And this is a limit of a quotient.
So your first temptation is to think the limit of a quotient the quotients the
limits. But what's the limit of the denominator?
The denominator is a polynomial. Polynomials are continuous.
So I can evaluate the limit of the denominator just by plugging in one.
And if I plug in one, I get one minus two plus one.
That's zero. The limit of the denominator is zero.
So I can't simply say that the limit of the quotient is the quotient of the
limits in this case. Because the limit of the denominator is
zero. And that limit law does not apply in this
case. What am I going to do?
Well I could also notice the denominator factors.
X squared minus 2X plus one, I can rewrite that.
[SOUND] X squared minus 2X plus one is X minus one squared.
[SOUND] So instead of evaluating this limit, I could try to look at this limit,
the limit of X plus two over X minus one squared as X approaches one.
Now what do I know? The numerator is X plus two and if X is
close to one, I can make X plus two close to three.
What about the denominator? A number close to one, minus one squared,
is a number close to zero. But not just a number close to zero.
It's a positive number close to zero. So let me write this down.
Alright. The numerator [SOUND] is about three.
[SOUND] And the denominator [SOUND] is about what?
Well, it's some small but positive number.
[SOUND]. And what happens if I take a number,
like, near three, and divide it by a number which is small and positive?
That number can be as big as I like. So I can make this quantity as large as I
like, if I make x close enough to one. Because I can make the numerator close to
three, and I can make the denominator as close to zero and positive as I like.
And if I take a number close to three and divide it by a number close to zero, I
can make a number as large as I want. So this limit [SOUND] is equal to
infinity. The homework includes even trickier
situations. For instance, limit problems where x
approaches negative infinity, instead of infinity.
If you get stuck I encourage you to contact us.
We're here to help you.